Multicast Routing 2

8/12/2019 Multicast Routing 2

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Multicast Routing

Algorithms


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Multicast Routing Algorithms 2

Outline

Introduction to Multi-point Communication

Three approaches to Multi-cast Routing

Steiner Tree Heuristics The MZQ algo

The SCTF algo

The Virtual Trunk algo for dynamic routing

The BSMA algo.

The KPP algo


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Multipoint Communication

Concept: Single Source, Multiple Destinations,

Duplication only at branch points.

Present Day Support: Communication satellites.

e-mail lists, internet news distribution.

Tomorrow's multimedia applications require:

efficient use of bandwidth.

near simultaneous delivery.


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Applications: Multicast & Multi-point

One to Many

Video Distribution

Wide scale Information

dissemination.

Many to Many

Video Conferencing

Computer Supported

Common Work.

Distributed interactive

simulation.

Large scale distributed

(super)computing.

Distributed Games


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Semantics of Multi-point Communication

Reliability

are different reliability models required for different classes of

applications?

Allowing dynamic join and leave session has to be receiver controlled.

Addressing

how to address groups at each level?

Whether and how to identify groups in layers above the IP layer?

Directionality

one-to-many or many-to-many.

are the transmitters a subset of the receivers?


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Multipoint Routing algos

Performance Metrics

Quality of a tree is judged according to the following three

dimensions

Low Delay:

End to end delay between source and receiver relative to theshortest unicast path delay.

Low Cost :

Cost of total bandwidth consumption

Cost of tree state info

Light Traffic Concentration :

Maximum number of flows on a unidirectional link.

How evenly the routes are distributed.


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Routing Algorithms

All multi-point services use some kind of a distribution tree.

Multicast trees can be

Shared across sources. (shared trees)

Only one tree needs to be established for each group, which is

shared by all the sources within that group.

Source specific. (shortest path trees).

A shortest path tree rooted at each sending node needs to be

established


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SOURCE BASED MULTIPOINT ROUTING

The Technique.

A Source Rooted Shortest Path Tree (SRSPT) algo:

Computes the shortest paths between the source and each of the

receivers within the group.

Eliminates duplicate data copies on common links.

Maintains one SRSPT per sender.

Concept:All receiving nodes compute path towards the source

independently.

Used by:current day IP multicast protocols as applications are still

small scale.

local area.


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SHARED TREE APPROACH OF MULTIPOINT ROUTING

Characteristics of Steiner Tree based algorithms.

The Minimum Steiner Tree: The minimal cost subgraph

spanning a given subset of nodes in a graph.

The Steiner Tree problem is NP-complete. finding the minimum steiner tree in a graph has exponential cost.

The tree designed is undirected.

solution feasible only for symmetric links.

Monolithic algorithm.

has to be run each time group membership changes.


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Characteristics of Steiner Tree based algorithms.

The SMT defines an absolute limit on the minimum tree

cost to serve as a reference for gauging the cost-optimality

of heuristic alternatives.

The SMT for all members of a multicast group is the same

irrespective of the role of sender or receiver.

only one state entry needs to be maintained per group.

it scales well for larger groups.

The SMT may have unbounded delay.

Worst case maximum end-to-end path length of a SMT can be the

longest acyclic path within the graph.


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An example of a minimum Steiner Tree

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Mcast group members

Relay Nodes

AG = 6AE = 5AD = 3AC = 3AI = 8

Shortest Paths from Atotal distribution cost = 16

*

AG = 8AE = 5AD = 3AC = 3AI = 8

KMB Tree paths from Atotal distribution cost=13


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Characteristics of Core Based Tree algorithms.

Concept:

Use the shortest Path Tree rooted at a node in the center of the

network

Steps: Choose an optimal center for the group. Multiple cores can be used

for better fault tolerance & delay characteristics.

Group members send a join message to the center.

Intermediate nodes mark interface from which the multicast info is

received and forward it to the center.

Choose the center to:

minimize max/avg delay for all members on the tree.

Minimize the sum of tree-link costs.


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An example of a core based tree.

{A, B, C} = multi-cast group members

A

B

C


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Advantages of Core Based Tree algorithms

Work well with multiple senders/receivers

state information is stored per group, therefore scalable.

Receiver based approach.

Supports dynamic group membership with relativeease.

Suitable for sparsely distributed receivers.

SPTs will not have many common links.

Do not have the unbounded delay problems of SMTs.

Simple to implement

used as the basis of PIM and of The CBT interdomain

Routing Protocol.


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Disadvantages of Core Based Tree algorithms

Incur extra delay as compared to the RPF approach.

Suffer from traffic concentration on links converging

towards the center.

Choosing the optimal center is an NP complete problem.

Locating the center requires complete knowledge of the

network topology.


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MULTIPOINT ROUTING

TradeOffs between algos

Any single tree cannot achieve Minimal Cost and Minimal

Delay both.

Shortest Path TreesMinimize delay at expense of Cost.

Steiner Minimal TreesMinimize cost at expense of Delay.

Between thesespectrum of different types of trees offering

different tradeoffs.

Different strategies to place the routes results in different

degrees of traffic concentration.


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MULTIPOINT ROUTING

Ideals

Ideally multicast routing algorithms should

Compute trees with the desired cost and delay

characteristics.

Adapt to dynamic group behavior. Algorithm should be incremental (like CBT) instead of

monolithic (like SMT).

Maintain properties of the original route.

Not perturb ongoing data transfers.Be receiver driven.


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STEINER TREE HEURISTICS

Problem Formulation

Given graph G= (V, E, c)

V= Set of vertices

E= Set of edges.

c= Cost function c: EZ0+ ( EdgesNon Negative Integers)

Z-Vertices: Set of Terminals (sometimes referred to as M)

S-Vertices: Set of non-terminals

TO: Initial tree = {s}.

Q : Priority Queue of vertices in the tree.

Vt: Vertices in the tree.

At: Edges in the tree.


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Pruned Dijkstra Heuristic (PDH)---Networks v17 `92

Take an arbitrary node as source.

Find the single source shortest path tree T for graph G

using Dijkstras algo. Delete from T, all S-vertices of degree 1.


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Dijkstras Shortest Path Algorithm.

Begin.

vV,

add v to set U,

initialize Distance(v) = cost(s, v) Distance(s) = 0; Remove s from U.

while U is not empty do

vany member of G with minimum distance.

Remove v from U. For each neighbor w of v, do

if member(w, U)

distance(w) = min(distance(w), cost(w, v) + distance(v) );

Stop.


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Matsuyamas Minimum Cost Path Heuristic (MPH)

---Math Japonica v24

Begin

T1: subtree of G containing one arbitrarily chosen Z vertexi .

k = 1;

Zk={i}.

Determine a Z-vertexiZ - Zkclosest to Tk

Construct a tree Tk+1by adding the minimum cost path from Tk toi

k = k+1.

If k < p go-to step2.

If k = p, output Tpas the solution

Stop.


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Matsuyamas Minimum Cost Path Heuristic

B

A

D E

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Tree built so far

Next Terminal

to be added to the tree

Idea:For each iteration while M is not empty--Pick up that node from M which closest to the tree built so far.

Data Structure Needed: All pair shortest paths (Floydd Warshalls algo O(n3)a)


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KMB - A Fast Algo for Steiner Trees.---Acta Informatica 1981

Output A Stiener Tree Thfor G and the Z-vertices.

Step 1: Construct a complete directed distance graph G1=(V1,E1,c1) fromG and Z.

Step 2: Find the minimum spanning tree T1of G1. (pick any to break ties)

Step3: Construct a subgraph GSof G by replacing each edge in T1by its

corresponding shortest path in G. (break ties arbitrarily).

Step 4: Find the minimum spanning tree TSof GS (break ties arbitrarily).

Step 5: Construct a Steiner tree THfrom TSby deleting edges in TSif

necessary, so that all the leaves in TH are Steiner points.

Worst case time complexityO(|S||V|2).

Costno more than 2(1 - 1/l) *optimal cost

where l= number of leaves in the steiner tree.


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Working of KMB

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Working of KMB

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Destination Nodes

Intermediate Nodes


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Multicast Tree Generation AlgorithmsThe MZQ Algorithm for multicasting in all optical networks

-- Malli, Zhang, Qiao

Limited wavelength conversion : every node is capable of converting

an input wavelength to only a subset of output wavelengths.

Sparse wavelength conversion : an input wavelength can be converted

to any output wavelength, but only a few nodes posses this capability.

Sparse Splitting : only a fraction of nodes can forward as many copies

as needed, and the rest of the nodes have no splitting capability.

MZQ assumes there are always enough wavelengths on each link.

Constructs multi-cast trees based on splitting capability of the nodes.

Nodes without splitting capability can have at-most one child in the

tree.


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Multicast Tree Generation AlgorithmsThe MZQ Algorithm: Routing


Algorithm maintains three sets of nodes

V:nodes in the tree through which the tree can grow .

(nodes with splitting capability).

V`:nodes in the tree through which the tree cannot grow.(nodes without splitting capability)

UV: set of Terminals not included in any tree so far.

Pick that node from UV which is nearest to the tree

Include as many destinations as possible in one multicastingtree.

For nodes not included in the preceding tree(s), algorithm called

recursively to construct another multicasting tree.


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Multicast Tree Generation AlgorithmsThe MZQ Algorithm: Wavelength Assignment


PerformanceMetrics: Number of Wavelengths

Total amount of Bandwidth (Total number of Channels)

Two counters maintained on each link

Ihighest wavelength index being used.

Nnumber of wavelengths being used.

Unlimited wavelengths on each link.

First-Fit algorithm used for wavelength assignment.


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The MZQ Algorithm

Multicasting forest in a NSF-NET like network

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Source

Node with full Splitting Capability

Node with no splitting capability


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Multicast Tree Generation AlgorithmsThe MZQ Algorithm for multicasting in all optical networks


Results:

The bandwidth savings from using multicasting saturate at

50%.

Multicasting reduces number of wavelengths required byas much as 60%.

Even when the network does not have any nodes that have

the splitting capability, multicasting reduces the

bandwidth consumed by 43% to 45%. No more than 75% of the nodes need to have the splitting

capability to obtain the same effect as having the splitting

capability in all the nodes.


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Multicast Tree Generation AlgorithmsSCTF-Algo (Selective Closest Terminal First)

S. Ramanathan, ---IEEE Infocom 1996

Initially Tree T = {source}.

Repeat until M is empty

Extend one branch from T to a terminal in M.

remove that terminal from M. Stop

Vertices in Tree are maintained as a priority queue with

priority(source) > priority(terminals) > priority(non-terminals).

Bin B holds first vertices of the queue.

Choose the path of least cost from all vertices in B to all

non-terminals not in the Tree.


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Multicast Tree Generation AlgorithmsSCTF-Algo: Formal Description.

---IEEE Infocom 1996

Init Q{s}, Vt{s}, At{ }.

While M not empty do

Bfirst min( , |Q| )vertices in Q.

Initialize PATH to any path from B to M.

For each vin B do

for each min M do

if cost( shortestPath( v, m) < cost (PATH) PATHP.

Branchsubpath ( w z ) : only wis in Vt.

Insert vertices in Branch into Q. VtVt{vertices in Branch} , AtAt{edges in Branch}.

MM { terminals in Branch}.

Return T.


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Multicast Tree Generation AlgorithmsSCTF-Algo: Subsuming three other algorithms


PDH, MPH, and KMB are special cases of the R-algo.

= 1. SCTF equivalent to PDH. = |M|+1 & Ignore non-terminals in B.

SCTF equivalent to KMB.

= |V|. SCTF equivalent to MPH.

As

from 1 to n, Tree_Cost

, and Running_Time

. Running_Time linearly.

Tree_cost very rapidly.

Low = good operating point.


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Multicast Tree Generation AlgorithmsR-Algo. Performance Characteristics


Running Time of R-Algo : O(m2+ e)

Assumption: Shortest Paths from every vertex to every

terminal are available. ( takes O( m. n. log(e) ) time )

Performance Guarantee

A= max{ A(I)/OPT(I)}

Tree cost 2 . m. Optimal_Cost

m= MAX[u, v] max(cost(u,v), cost(v, u))/ min(cost(u,v), cost(v, u))m = 1 for symmetric graphs.

A= O (m).


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Multicast Tree Generation AlgorithmsFeatures of the SCTF-Algo


Controlling knob enables use of the SCTF-algo for both

Delay Optimization. ( = 1) Cost optimization. ( = n)

Advantageous for MultiMedia applications

select the right tradeoff & operating point to accommodate the

differing requirements of voice, video and data.


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Multicast Tree Generation AlgorithmsVTDM - A Dynamic Multicast Routing Algorithm

H.C.Lin & S.C. Lai ---IEEE Infocom 1998

Problem Formulation..

Source node s.

Sequence of requests R = {r1, r2, ... rm}

Each request rieither adds a destination node to

or removes a destination from the multicast group.

The DMRP: Find a sequence of multicast trees

{Ti, i = 1 .. m} such that certain overall costis minimum.

This does not allow re-routing of existing connections as

the sequence of requests proceeds.


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Cost Modelling

w(e, i):Cost of using wavelength ion edge e.

Infinite if iis not available on edge e.

cv(p , q):Cost of wavelength conversion at node v, frompto q.

Infinite if p cannot be converted toq at node v.

If p = q, then cv(p , q)is zero.

C(T) = v Tw(p(v), v), (v)) +

v T-{s}C p(v)((p(v)), (v))

where p(v) means parent of v in the tree.


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The VTDM algo

Concept of a Virtual Trunk

---Infocom 1998

A virtual trunk is a tree of the underlying graph.

Spans nodes which have the greatest probability of being a

part of the multi-casting tree. Used as a template for building the multicasting tree.

Nodes which have a greater number of shortest paths

passing through them, have a greater probability of being a

part of the multi-cast tree. Weight W(vi) of vertex vi= number of shortest paths

passing through vi.


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The VTDM algo

Building the Virtual Trunk---Infocom 1998

Find the shortest paths for all pairs of nodes in G.

Assign weights to the vertices in G.

Find the set of trunk nodes F.

Construct a complete graph for the set of trunk nodes.

Find the minimum spanning tree for the complete graph.

Convert edges in min. span. Tree back to the corres

shortest paths in graph G.

Run the minimum spanning tree algo and remove

unnecessary nodes and links to obtain the virtual trunk.


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The VTDM algo

The VTDM routing algorithm

---Infocom 1998

Build the virtual trunk.

Adding a node to the multicast group

establish shortest route from the node to the virtual trunk. is

established.

Route along virtual trunk to source node also established if not yet

there.

Add node to the multicast group.

Removing a node from the multi-cast group First remove the node from the multicast group.

If it is a leaf node, remove the node from the tree.

Prune the excess branch, if the node did not have any downstream

nodes.


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The VTDM algo

Node Addition (adding node B)---Infocom 1998

Step1:

If node B on the virtual trunk, denote it as node A & go to step2.

Else, find the shortest route from node B to the virtual trunk.

Add portion of the shortest route not yet included in the multicast

tree to the multicast tree.

Let node A be the node on the virtual trunk which attaches node B

to the virtual trunk via the selected shortest route.

Step2:

If node A is already on the multicast tree go to step 3. Else add portion of route from node A to source node that has not

yet been included in the multicast tree to the multicast tree.

Step3:

Add node B to the multicast group.


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The VTDM algo

Node Removal (removing node B)---Infocom 1998

Step1:

Remove node B from the multicast group.

Step2:

If node B has downstream nodes the procedure is done.

Else, if node B is a leaf, remove node B and its upstream link to

the multicast tree.

Step3: If the upstream node of node B is in the multicast group, the

procedure is done.

Else denote this node as node B and go to step 2.


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The VTDM algo

Simulation Results---Infocom 1998

Mean Inefficiency = TreeCost using AlgoA/ TreeCost using AlgoB.

KMBis taken as the reference algorithm.

VTDM compared against dynamic greedy(DG), Shortest Path (SP).

Mean Inefficiency versus Number of nodes

significant improvement over SP, better than DG.

Mean Inefficiency versus Size of multicast group.

significant improvement over SP, better than DGfor large grps.

Max Degree of nodes in the multicast trees. (no. of data copies).

Much lesser degree than SP, less than DGalgorithm.


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Multicast Tree Generation AlgorithmsBSMA - Bounded Shortest Multicast Algorithm

Zhu, Parsa & Aceves---IEEE Infocom 1995

Problem :

Minimize the tree cost.

Guarantee all delays are less than predetermined bounds.

Feasible region :

the set of all delay bounded Steiner trees.

Steps:

Construct minimum delay steiner tree T0using

Dijkstras shortest path algorithm

Refine T0iteratively for lower cost while staying within

feasible region.


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Multicast Tree Generation AlgorithmsBSMA - Definition of the cost function.


Utilization Driven Cost

Minimizes sum of link costs along the path.

Congestion Driven Cost

Minimizes maximal link cost requirement along paths.

Link cost function

Cost of the link associated with the utilization of the link.

Link delay function

Queuing, Transmission, and Propagation delays on the link.

Destination Delay Bound Function (DDF)

Upper bound to the delay along path from the source to each of the

destinations.


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Multicast Tree Generation AlgorithmsBSMA - Refinement of the tree for lower costs.


Path Switching:

refinement of Tj to Tj+1.

Choosing a path p to be taken out of Tj.

Tj = Tj1+ Tj

2p

Choosing the new path ps in G not in Tjthat replaces the

path to be deleted from Tj.

Tj+1 = Tj1+ Tj

2ps. Tj+1is delay bounded.

l G l h


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Multicast Tree Generation AlgorithmsBSMA - Refinement of the tree for lower costs.


From Tjget Tj/.

Tj/has the source, all destination nodes, and all relay nodes of

degree more than 2.

Tj/. Edges of Tj

/are called super_edges.

All nodes between the two end_nodes of a super_edge are relay

nodes of degree 1.

Every super_edge represents a candidate path in Tj for switching.

**

M lti t T G ti Al ith


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Multicast Tree Generation AlgorithmsBSMA - Algorithm Details


Initially allsuper_edgesare unmarked.

Step1:Among all unmarkedsuper_edges, BSMA selects the

super_edgePhwith the highest path cost.

Exchange it with anothersuper_edgeof lesser cost, such that resulting

tree is delay bounded.. One of the two cases must happen:

The delay bounded shortest path is the same as Ph.

Mark that super edge. Go to Step1.

The delay bounded shortest path is a path other than Ph.

Do The replacement.

Unmark all super_edges.

Go to Step 1.

Stops when all super edges are marked.



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Multicast Tree Generation AlgorithmsBSMA - Algorithm Details


BSMA incrementally calculates k shortest paths between subtrees Tj1

and Tj2.

K is determined only after a shortest path is found which has resulted

in a delay bounded tree. So the shortest path incremental construction

stops when one of the following two conditions is satisfied. The shortest path found does not result in the new tree violating the

delay bound.

The shortest path found has equal path length to the one just

deleted.

Dijkstras algo is extended to construct shortest path between twosubtrees instead of two nodes

A pseudo source node s is connected to all nodes in Tj1 and a

pseudo destination node d is connected to all nodes in Tj2.

The shortest path algo starts from s and ends at d.


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The BSMA algo

cont

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A B

CD

Th BSMA l


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The BSMA algo

Greedy Path switching

---IEEE Infocom 95

Gain = cost reduction after a round of path switching

if c = cost of Tj and c_prime = cost of Tj+1, then

gain = c - c_prime.

BSMA computes gains of all pairs of possible path

switchings in Tj and then selects one with the maximum

gain.

BSMA continues the greedy switching and terminateswhen the maximum gain is zero.

The time complexity of this greedy approach is more.


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The BSMA algo

Simulation Results

Time complexity of BSMA O(kn3log(n)).

Using the tightest possible delay bound, as determined by the

min. delay tree, the cost of the BSMA tree is substantially better

than the cost of the min. delay tree.

A Range of min. cost solutions can be obtained between the two

extremes of the KMB and the min. delay solution.

Compared to KMB the relative quality of results improve with

the number of destinations in the multicast group.

A tighter bound results in a larger value for k, and henceincreases the computation time required by the algo, a slightrelaxation of the bound often results in considerably fewercomputations.



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Multicast Tree Generation AlgorithmsKPP algorithm for a delay constrained Steiner Tree

Kompella, Pasquale, Polyzos ---IEEE Infocom 1995

Problem :

Minimize the tree cost. is minimized.

Bounded end-to-end delay.

Features:

Edge_Cost and Delay are different functions.

Delay constraints are on individual path delay.

Assumption:

Source has all the info necessary for tree construction.

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Multicast Tree Generation Algorithms


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Multicast Tree Generation AlgorithmsKPP algorithm for a delay constrained Steiner Tree


Constrained Cheapest Path between v and w

Least cost path between vand wthat has delay less than .

Cost of such a path is PC(v, w).

Delay on this path is PD(v, w). Closure graph G/

A complete graph on the nodes in N, with PC(v, w) as edge

costs PD(v, w) as the edge delay.

To compute Closure graph G/

Calculate all-pair-constrained-cheapest-paths using Floyds

algorithm. (is bounded, so possible in poly-time)



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Multicast Tree Generation AlgorithmsKPP algorithm details


Cd(v, w) = minuV {Cd-D(u, w)(v, u)+ C(u, w)}

Cost of cheapest path from vto wwith delay exactly d.

PC(v, w) = min d


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Multicast Tree Generation AlgorithmsKPP algorithm details


Compute the closure graph G/of G.

Construct a constrained spanning tree T/of G/, using one ofthe two selection functions fCor fCDas the selection

function.

Expand the edges in the constrained spanning tree T/ intothe constrained cheapest paths they represent, and remove

any loops that may be caused by this expansion.



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Multicast Tree Generation AlgorithmsKPP algorithm - Two Source Based Heuristics


CSTCD

Tries to choose low cost edges but modulates the choice

to pick edges that maximize the residual delay.

This increases the chances of extending the path alongthis edge, and beyond to another destination.

Has a tendency to optimize on delay.

May find trees with delay far lower than at the

expense of added cost to the tree. Uses fCD= if P(v) + D(v, w) <

fCD= infinity otherwise

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Multicast Tree Generation AlgorithmsKPP algorithm - Two Source Based Heuristics


CSTC

minimizes fC.

Constructs cheapest tree possible, while remaining

within delay bounds. Minimizes cost, without unduly minimizing delay.

fc = C(v, w) if P(v) + D(v, w)

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